HEALPix (sometimes written as Healpix), an acronym for Hierarchical Equal Area isoLatitude Pixelisation of a 2-

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

, is an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

for

pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the ...

isation of the 2-sphere and the associated class of

map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longit ...

s. The pixelisation algorithm was devised in 1997 by Krzysztof M. Górski at the Theoretical Astrophysics Center in Copenhagen, Denmark, and first published as a preprint in 1998.

Projection and pixelisation

The HEALPix projection is a general class of spherical projections, sharing several key properties, which map the 2-

to the Euclidean plane. Any of these can be followed by partitioning (pixelising) the resulting region of the 2-plane. In particular, when one of these projections (the H=4, K=3 HEALPix projection) is followed by a pixelisation of the 2-plane, the result is generally known as the HEALPix pixelisation, which is widely used in

physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...

for maps of the

cosmic microwave background In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...

. This pixelisation can be thought of as mapping the sphere to twelve square facets (diamonds) on the plane followed by the binary division of these facets into pixels, though it can be derived without using the projection. The associated software package HEALPix implements the algorithm. The HEALPix projection (as a general class of spherical projections) is represented by the keyword ''HPX'' in the FITS standard for writing astronomical data files. It was approved as part of the official FITS World Coordinate System (WCS) by the

International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreac ...

FITS Working Group on April 26, 2006. The spherical projection combines a cylindrical equal area projection, the Lambert cylindrical equal-area projection, for the equatorial regions of the sphere and a pseudocylindrical equal area projection, an interrupted

Collignon projection The Collignon projection is an equal-area pseudocylindrical map projection first known to be published by Édouard Collignon in 1865 and subsequently cited by A. Tissot in 1881. For the smallest choices of the parameters chosen for this projecti ...

, for the polar regions. At a given level in the hierarchy the pixels are of equal area (which is done by bisecting the

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

in the case of the H=4, K=3 projection) and their centers lie on a discrete number of circles of latitude, with equal spacing on each circle. The scheme has a number of mathematical properties which make it efficient for certain computations, e.g.

spherical harmonic In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...

transform Transform may refer to: Arts and entertainment * Transform (scratch), a type of scratch used by turntablists * ''Transform'' (Alva Noto album), 2001 * ''Transform'' (Howard Jones album) or the title song, 2019 * ''Transform'' (Powerman 5000 album ...

s. In the case of the H=4, K=3 projection, the pixels are squares in the plane (which can be inversely projected back to quadrilaterals with non-geodesic sides on the 2-sphere) and every vertex joins four pixels, with the exception of eight vertices which each join only three pixels. The latitude of transition between equatorial-orthogonal and polar-convergent longitude lines has been selected to allow the folding of the projection into a perfect cube — "cubing the sphere"; indeed in this way the Arctic Circle becomes a square.

Usage and alternatives

The pixelisation related to the H=4, K=3 projection has become widely used in cosmology for storing and manipulating maps of the

Gaia mission ''Gaia'' is a space observatory of the European Space Agency (ESA), launched in 2013 and expected to operate until 2025. The spacecraft is designed for astrometry: measuring the positions, distances and motions of stars with unprecedented preci ...

uses HEALPix as the basis for source identification. An alternative hierarchical grid is the Hierarchical Triangular Mesh (HTM). The pixels at a given level in the hierarchy are of similar but not identical size. The scheme is good at representing complex shapes because the boundaries are all segments of circles of the sphere. Another alternative hierarchical grid is the

Quadrilateralized Spherical Cube In mapmaking, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area polyhedral map projection and discrete global grid scheme for data collected on a spherical surface (either that of the Earth or the celestial sphere). ...

. The 12 "base resolution pixels" of H=4, K=3 HEALPix projection may be thought of as the facets of a

rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...

. The H=6 HEALPix has similarities to another alternative grid based on the icosahedron.What is the best way to pixelize a sphere?
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References

External links

Official implementation
with many languages support (C, C++, Fortran90, IDL, Java and Python) for resolutions up to 0.4 mas (

milliarcsecond A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...

)
Java port
of original Fortran code by Nikolay Kuropatkin, supporting resolutions up to 0.3 arcsec
Java port
optimized to use RangeSet, very good for high resolutions
healpy
Python wrapper
astropy-healpix
BSD-licensed HEALPix for Astropy
healpix.cxx
c++ code to transform healpix coordinates
JavaScript port
convert between lonlat and HEALPix coordinates in JavaScript
Typescript healpix
An implementation of HEALPix in JavaScript / TypeScript
Healpix.jl
Healpix library written in Julia {{DEFAULTSORT:Healpix Map projections

Projection and pixelisation

Usage and alternatives

See also

References

External links